Graphene and carbon nanotubes in transistors, diodes and field emission devices

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D

IPARTIMENTO DI

F

ISICA “

E

.

R

.

C

AIANIELLO”

THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHYLOSOPHY IN PHYSICS

XIII CYCLE - II SERIES (2012-2016)

G

RAPHENE AND

C

ARBON

N

ANOTUBES IN

T

RANSISTORS,

D

IODES AND

F

IELD

E

MISSION

D

EVICES

L

AURA

I

EMMO

S

UPERVISOR

PROF ANTONIO DI BARTOLOMEO

C

OORDINATOR

P

ROF

C

ANIO

N

OCE

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To my children Aurora and Ivan

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i

Introduction

Since their discovery, graphene and carbon nanotubes have been playing an important role in nanoscience and nanotechnology thanks to their extraordinary physical and chemical properties.

Graphene, the two-dimensional layer of sp2_{carbon atoms discovered in 2004 [1], is an}

excellent material for electronic devices for its linear dispersion relation with electrons behaving as massless Dirac fermions [2], high electron mobility [3], great electric current carrying capacity [4], capability of being tuned from p-type to n-type doping by the application of a gate voltage, high thermal conductivity [5], record mechanical strength [6], resilience to high temperatures [7] and humidity [8], structural flexibility, resistance to molecule diffusion and chemical stability. Because of these exceptional properties, graphene has stirred a lot of interest in the scientific community for basic science and for technological applications, and it is considered a potential breakthrough in terms of carbon-based nanoelectronics.

With silicon-based electronics tending towards its scaling limits, the semiconductor industry is looking for the next switch which can replace the silicon field-effect transistor [9] and graphene can be a possible alternative for silicon. Graphene-based field-effect transistors (GFETs) [10] combine an ultra-thin body suitable for aggressive channel length scaling [11], with exceptional electronic properties [12]. However the development of graphene-based electronics is limited by the quality of contacts between the graphene and metal electrodes [13-15] that can significantly affect the electronic transport properties of the devices [16]. For such reason it becomes of fundamental importance to characterize metal/graphene interfaces at the contacts.

Further, due to its important advantage of being naturally compatible with thin film processing, graphene is easy to integrate into existing semiconductor device technologies. The graphene-silicon (Gr/Si) heterojunction is one of the simplest conceivable device in a hybrid graphene-semiconductor technology and it offers great opportunity to study the physics occurring at the interface between a 2D and a 3D material, as well as between a zero and a definite bandgap system, and can be a convenient platform to investigate electronic properties and transport mechanisms. Nonetheless, the Gr/Si junction has already been demonstrated as a rectifying or a barrier-variable device, a photovoltaic cell [17-19], a bias-tunable photodetector [20-22], a chemical-biological sensor [23-25] and it is gaining interest

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from the semiconductor industry also for the potentiality to replace ultra-shallow doped junctions in modern complementary-metal-oxide-semiconductor (CMOS) technologies.

Carbon nanotubes (CNTs), for their very high aspect ratio (diameter in the nanometer scale and length of several microns), extremely small radius of curvature, unique electric properties, high chemical stability and important mechanical strength [26-29], have been considered extraordinary elements to realize field emission devices, since their discovery in 1991 [30]. Field emission (FE), which involves extraction of electrons from a conducting solid by an external electric field, is at the basis of several technological applications. Nowadays, CNT based field emitters are used in vacuum electronics to produce electron sources [31], flat panels [32], X-ray sources [23,24], and microwave amplifiers [35], exploiting a low-threshold electric field and large emission current density. As CNTs, also graphene has high potentiality for FE applications, due to its high aspect ratio (thickness to lateral size ratio) and a dramatically enhanced local electric field is expected at its edges.

The aim of this PhD thesis was to study electronic properties and transport mechanisms of graphene and carbon nanotubes through an extensiveelectrical characterization of field-effect transistors, diodes and field emission devices based on these materials.

The thesis is organized as follows.

In the first chapter, we describe the physical and electronic structures of graphene and carbon nanotubes in order to understand the transport properties and performance of studied devices. Moreover, we report a brief introduction to the optical properties of graphene and the field emission theory.

In the second chapter, we present the fabrication of graphene based field effect transistors in bottom and side gate configuration, and we perform an intense electrical characterization by measuring transfer and output characteristics. In particular, we study the physical effects due to the contact resistance between graphene and different metal electrodes. We discuss the effects of room temperature vacuum degassing and of low-energy electron beam irradiation on GFETs. Finally, we study the side-gating effect, the gate leakage in all-graphene devices and the field emission from graphene.

In the third chapter, we study a new-concept of Gr/Si photodiode made of a single-layer graphene transferred onto a matrix of nanotips patterned on n-type Si wafer. Through an extensive characterization, we estimate the relevant junction parameters. In particular, we analyze the effect of the tip geometry on the Schottky barrier height and on the photoresponse.

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In the fourth chapter, we describe the transport characteristics in a wide temperature range and the field emission properties of buckypapers obtained from aligned carbon nanotubes. We report the study of the long-term stability of the field emission current and finally, we analyze the effect on the emitted current due to in plane applied currents in the buckypaper.

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Bibliography

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[23] Kim H Y, Lee K, McEvoy N, Yim C and Duesberg G S 2013 Nano Lett. 13 2182–2188. [24] Singh A, Uddin A, Sudarshan T and Koley G 2014 Small 10 1555-1565. [25] Fattah A, Khatami S, Mayorga-Martinez C C, Medina-Sánchez M, Baptista-Pires L and Merkoçi A 2014 Small 10 4193–4199.

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6

Chapter 1

Graphene and carbon nanotubes:

theoretical and experimental background

1.1 Introduction

Carbon is one of the most versatile elements in the periodic table in terms of the number of compounds it may create, mainly due to the types of bonds it may form (single, double, and triple bonds) and the number of different atoms it can join in bonding. It can form a broad variety of architectures in all dimensions, both at the macroscopic and nanoscopic scales.

During the last 25 years, brave new forms of carbon have been unveiled. The family of carbon-based materials now extends from C60 to carbon nanotubes, and from old diamond and graphite to graphene. The properties of the new members of this carbon family are so impressive that they may even redefine our era.

Under standard conditions (ordinary temperatures and pressures), the stable form of carbon is graphite. Graphite is a famous lubricant, an electrical (semimetal) and thermal conductor, and reflects visible light [1].

Graphite is a three-dimensional crystal made of stacked layers consisting of sp2_hybridized carbon atoms; each carbon atom is connected to another three making an angle of 120° with a bond length of 1.42 Å. This anisotropic structure clearly illustrates the presence of strong σ covalent bonds between carbon atoms in the plane, while the π bonds provide the weak interaction between adjacent layers in the graphitic structure. The σ-bonds, that have the electrons localized along the plane connecting carbon atoms, are responsible for the great strength and mechanical properties of graphene and carbon nanotubes [2,3], while the 2p electrons, that are weakly bound to the nuclei and hence relatively delocalized, are the ones responsible for the electronic properties of graphene and carbon nanotubes [1,4].

The objective of this chapter is to describe the physical and electronic structures of graphene and carbon nanotubes in order to understand the electronic properties and transport mechanisms in the studied electronic devices. Briefly, we report the optical properties of

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graphene, that are important for the analysis of graphene-based optoelectronic devices. Finally, we introduce the field emission theory, whose properties and applications are currently the subject of a very active research field.

1.2 Graphene

Isolated graphene was discovered in 2004 by Geim and Novoselov [5,6], who made graphene accessible with a technique as simple as the mechanical exfoliation.

Graphene is a one-atom thick, planar layer consisting of sp2_{hybridized carbon atoms}

arranged in a 2D hexagonal honeycomb. The planar honeycomb structure of graphene has been observed experimentally and is shown in Fig. 1.1 [4].

FIGURE 1.1: Remarkable transmission electron aberration-corrected microscope (TEAM) image of graphene vividly showing the carbon atoms and bonds in the honeycomb structure.

Graphene can be considered the mother of three carbon allotropes. As illustrated in Fig. 1.2, wrapping graphene into a sphere produces buckyballs, folding into a cylinder produces nanotubes, and stacking several sheets of graphene leads to graphite [7].

FIGURE 1.2: Two-dimensional graphene can be considered the building block of several carbon allotropes in all dimensions, including zero-dimensional buckyballs, 1D nanotubes, and 3D graphite.

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1.2.1 The direct lattice

Graphene has a honeycomb lattice shown in Fig. 1.3 using a ball-and-stick model [4]. The balls represent carbon atoms and the sticks symbolize the σ-bonds between atoms. The carbon–carbon bond length is approximately ac-c ≈ 1.42 Å.

The honeycomb crystal can be mapped to a triangular Bravais lattice with a basis of two atoms, which can be considered as made of two interpenetrating triangular sub-lattices, indicated as A and B in Fig. 1.3. These contribute a total of two π electrons per unit cell to the electronic properties of graphene. The primitive unit vectors as defined in Fig. 1.3are:

= √_{2 , 2 , =}3 √3_{2 , − 2 , (1.1)} with | | = | | = = √3 = 2.46 Å .

Each carbon atom is bonded to its three nearest neighbors and the vectors describing the separation between a type A atom and the nearest neighbor type B atoms as shown in Fig. 1.3 are:

=

√3, 0 , = − + = −2√3, − 2 , = − + = −2√3,2 , (1.2)

with | | = | | = | | = .

FIGURE 1.3: The honeycomb lattice of graphene. The primitive unit cell is the equilateral parallelogram (dashed lines) with a basis of two atoms denoted as A and B.

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1.2.2 The reciprocal lattice

The reciprocal lattice, that is the discrete Fourier transform of the direct lattice, satisfies the basic relation:

· _{= 1, (1.3)}

where K is the set of wavevectors that determine the sites of the reciprocal lattice points and

R is the Bravais lattice position vector [4].

In two dimensions, the primitive vectors of the reciprocal lattice (b1 and b2) are determined

from the primitive vectors of the direct lattice (a1 and a2):

= 2 _{( , ), = 2}( ) _{( , ), (1.4)}(− ) where R90 is an operator that rotates the vector clockwise by 90° and det is the determinant,

which geometrically is the area of the parallelogram formed by a1 and a2 and serves as a

normalization factor. From the rotation operator, it is evident that the reciprocal lattice primitive vectors are either normal or parallel to the direct lattice primitive vectors, corresponding to ai · bj = 2πδij , where δij is the Kronecker delta function.

The reciprocal lattice of graphene shown in Fig. 1.4 is also a hexagonal lattice, but rotated 90° with respect to the direct lattice. The reciprocal lattice vectors are (from eq. (1.4)):

= 2 √3 , 2 , = 2 √3 , − 2 , (1.5) with | | = | | = 4 /√3 .

FIGURE 1.4: The reciprocal lattice of graphene. The first Brillouin zone is the shaded hexagon with the high symmetry points labeled as Γ, M, and K.

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In the Brillouin zone, which is illustrated as the shaded hexagon in Fig. 1.4, there are three key locations of high symmetry. In Fig. 1.4, these locations are identified by convention as the Γ-point (point located at the center of the hexagon), the M-point (midpoint of the side of the hexagon) and the K-point (corner of the hexagon). The vectors describing the location of the points M and K with respect to the zone center are:

= 2 √3 , 0 , = 2 √3 , 2 3 . (1.6) There are six K-points and six M-points within the Brillouin zone. Sometimes a distinction is made between the K-point and K'_{-point (Fig. 1.4), particularly in the discussion of intervalley}

or interband electron scattering by lattice vibrations, but they are essentially equivalent for most purposes. The unique solutions for the energy bands of crystalline solids are found within the Brillouin zone and sometimes the dispersion is graphed along the high symmetry directions.

1.2.3 _{Electronic properties}

The electronic band structure of graphene is of primary importance because it is the starting point for the understanding of graphene’s solid-state properties and analysis of graphene devices and it is also the starting point for the understanding and derivation of the band structure of CNTs.

The band structure of graphene is shown in Fig. 1.5. This band structure was computed numerically from first principles and shows many energy branches resulting from all the π and σ electrons that form the outermost electrons of carbon [8].

There is a limiting technique for obtaining a satisfactory band structure, called the “tight-binding model”, that is in good agreement with experimental measurements or more sophisticated numerical ab-initio band structure computations. This model assumes that the outermost electrons are localized (i.e. tightly bound) to their respective atomic cores.

The assumptions belonging to the tight-binding formalism are [4]:

-_{Nearest neighbor tight-binding (NNTB) model: the wavefunction of an electron in any} primitive unit cell only overlaps with the wavefunctions of its nearest neighbours. The nearest neighbors of a type-A atom in the graphene lattice are three equivalent type-B atoms.

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- Electron–hole symmetry: a close observation of the ab-initio dispersion in the neighborhood of the Fermi energy (E = 0 eV at the K-point in Fig. 1.5) reveals that the π and π*_{branches have similar structure, at least for energies close to the E}

F. Within this

restricted range, the energy branches are approximately mirror images of each other. Since electrons are the mobile charges in the π*_{band and holes are the mobile charges in the π}

band, this approximation is called “the electron–hole symmetry”.

FIGURE 1.5: The ab-initio band structure of graphene, including the σ and π bands. The Fermi energy is set to 0 eV [8].

As a result of the NNTB stipulations, the dispersion relation E(k) is [4,9,10]:

( ) = 1 + 4cos √3₂ _{cos 2 + 4} _{2 , (1.7)} where γ , including the nearest neighbor overlap energy, is called “ the hopping energy” and it is often used as a fitting parameter to match ab-initio computations or experimental data. Commonly used values for γ range from about 2.7 eV to 3.3 eV.

Comparison of the NNTB dispersion, eq. (1.7), with ab-initio computations for the π bands shows good agreement (Fig. 1.6) with the strongest agreement expectedly at low energies (range within ±1 eV is sometimes considered reasonable) [11]. Much of the exploration of graphene and derived nanostructures such as CNTs has been focused on the low-energy properties.

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FIGURE 1.6: Comparison of ab-initio and NNTB dispersions of graphene showing good agreement at low energies (energies about the K-point). γ = 2.7 eV is used [11].

Fig. 1.7 shows the 3D plot of the NNTB dispersion throughout the Brillouin zone. The upper half of the dispersion is the conduction (π*) band and the lower half is the valence (π) band. The highest occupied state housing the most energetic electrons are at the K-points, as identified earlier, and the corresponding energy is formally defined as the Fermi energy (EF = 0 eV). The properties of electrons around the Fermi energy often determine the

characteristics of practical electronic devices.

Owing to the absence of a bandgap at the Fermi energy, and the fact that the conduction and valence bands touch at EF, graphene is considered a semi-metal or zero-bandgap

semiconductor, in contrast to a regular metal, where EF is typically in the conduction band,

and a regular semiconductor, where EF is located inside a finite bandgap. Under

non-equilibrium conditions (applied electric or magnetic fields) or extrinsic conditions (presence of impurity atoms), the Fermi energy will depart from its equilibrium value of 0 eV.

FIGURE 1.7: The nearest neighbor tight-binding band structure of graphene: the π and π* bands are symmetric with respect to the valence and conduction bands. The hexagonal Brillouin zone is superimposed and touches the energy bands at the K-points. The linear dispersion relation close to the K' and K points of the first 2D Brillouin zone gives rise to the “Dirac cones” as shown on the right.

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Expanding eq. (1.7)for k in the vicinity of K (or K'), yieldsa linear dispersion for the π and

π* bands near these six corners of the hexagonal 2D Brillouin zone [1,4,12]:

( ) = ħ | |, (1.8)

where:

= √_{2ħ , (1.9)}3

is the electronic group velocity, called “ the Fermi velocity”, and ħ is the reduced Planck’s constant. Graphene is thus highly peculiar for this linear energy-momentum relation and electron-hole symmetry. The electronic properties in the vicinity of these corners of the 2D Brillouin zone mimic those of massless Dirac fermions forming “Dirac cones” as illustrated in

Fig. 1.7. The six points where the Dirac double cones touch are referred to as the Dirac points. The electronic group velocities close to those points are quite high at ~ 106_{m/s, and within} the massless Dirac fermions analogy represent an effective “speed of light” (vF ~ c/300). This

behavior is responsible for much of the research attention that graphene has been receiving as platform for investigating the properties of the Dirac fermions and for perspective high-speed electronic applications.

Another important property of graphene is a special feature of the carrier wavefunction which leads to other unusual properties. Due to the two interpenetrating sub-lattices, A and B, carriers near the Dirac point K, can be described by a two component wavefunction [10]:

, ( ) = 1

√2 , (1.10) where θk = arctan(qx/qy) and q = k-K is the momentum measured from the Dirac point K.

This wavefunction has some interesting implications. For example, electrons along + kx and

- kx have orthogonal wavefunctions, so there is no probability of backscattering at 180 degrees

which favors mobility and enables near ballistic transport at room temperature. The mobility

µ, in

cm2_V-1_s-1_{, is defined both in a diffusive or ballistic regime as the ratio between the} conductivity

σ and the carrier charge density and is commonly used to characterize the

graphene structural quality:

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The conductivity (which for a two-dimensional system coincides with the so called “sheet conductance”) and the mobility of graphene depend on the microscopic scattering processes that occur in graphene at a given temperature [13-15].

The carrier density n sets the position of the Fermi level, EF, with respect to the Dirac

point. For ideal neutral graphene without free carriers, EF is located at the Dirac point, where

n ≈ 0 (n = 0 at T = 0 K). Graphene becomes an n- or p-type conductor when the Fermi level

shifts above or below the Dirac point and n corresponds to an excess of electrons or holes, respectively.

The relation between n and EF can be easily derived considering that the density of states

(DOS) in graphene depends linearly on the energy E. Formally, in two dimensions, the total number of states available between an energy E and an interval dE is given by the differential area in k-space dA divided by the area of one k-state (2π)2_{/Ω, where Ω is the area of the}

lattice. Mathematically, this is equivalent to [4,14]:

( )d = 2g _{(2 ) /Ω, (1.12)}d where the factor of two in the numerator is included for spin degeneracy and gz is the zone

degeneracy. There are six equivalent K-points, and each K-point is shared by three hexagons; therefore, gz =6·1/3=2 for graphene. To determine dA, let us consider a circle of constant

energy in k-space of radius k. The differential area obtained by an incremental increase of the radius by dk is 2πkdk. Therefore, the DOS, which is always a positive value or zero, is:

g( ) = ( )_{Ω =}2_π d_{d =}2_π d_d . (1.13) Substituting from eq. (1.8) yields a linear DOS appropriate for low energies:

g( ) = _{(ħ ) | | = | |, (1.14)}2 where βg is a material constant, βg ≈ 1.5·106 eV-2µm-2 and the absolute value of E is necessary

because energy can be either positive (electrons) or negative (holes). At the Fermi energy (EF = 0 eV), the DOS vanishes to zero even though there is no bandgap. This is the reason

whygraphene is considered a semi-metal in contrast to regular metals that have a largeDOS at the Fermi energy.

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The equilibrium electron carrier density n is [4,12]:

= g( ) (

∞

)d =2 _ħ ≈1 _ħ , (1.15) where f(E) is the Fermi–Dirac distribution function:

( ) =_{1 +} ₍ 1 _)/ , (1.16) and F1(EF/kBT) is the so-called “Fermi-Dirac Integral” and the last expression rigorously

holds at T = 0 K. Eq. (1.15) shows that the Fermi level changes as the square root of the carrier density:

= ℎ

2√ √ , (1.17) where the + and - sign correspond to n and p-type graphene, respectively).

The position of EF can be varied experimentally either by chemically doping the graphene

[16] or by inducing an excess of carriers by means of an electric field generated by an applied bias or a gate [17]. The possibility of controlling the position of the Fermi energy, the sign of the excess carries and hence the conductivity by doping or by a bias/gate is a remarkable feature of graphene and opens the possibility of a new class of electronic devices.

1.2.4 Graphene nanoribbons

Graphene nanoribbons are narrow rectangles made from graphene sheets and have widths on the order of nanometers up to tens of nanometers [4]. The nanoribbons can have arbitrarily long length and, as a result of their high aspect ratio, they are considered quasi-1D nanomaterials. GNRs can have metallic or semiconducting character. GNRs show a departure from the electronic properties of graphene sheets, most notably the opening of a bandgap due to the quantum confinement and edge effects. The opening of a bandgap is of great interest because it unlocks the potential of employing GNRs as transistors.

The band structure of GNRs can be computed numerically using first principles or tight-binding schemes [18]. A useful first-order semi-empirical equation capturing the width dependence of the bandgap Eg has a simple relation:

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= + , (1.18) where w (nm) is the width of the nanoribbons and w0 (nm) and α (eV·nm) are fitting

parameters. Experimentally extracted values of α range from 0.2 eV to about 1 eV [19,20]. Experimental and theoretical data suggest a w0 ≈ 1.5 nm. As the width of the nanoribbon

increases and exceeds about 50 or 100 nm, Eg vanishes and the band structure of GNRs

gradually returns to that of a 2D graphene sheet. Fig. 1.8 plots a set of experimentally extracted values for the bandgap confirming the inverse width dependence [19].

FIGURE 1.8: Experimentally extracted bandgap versus width for GNRs [19].

1.2.5 Transport properties

At the Dirac point, even at cryogenic temperatures when n and

σ should tend to zero, the

conductivity of graphene remains finite [12]. This is a consequence of the intrinsic properties of the 2D Dirac fermions, which set a limit on the minimum attainable conductivity. In short and wide strips ( width to length ratio W/L≫1 ) of ideal graphene, with no impurities or defects and for T → 0K, transport at the Dirac point is explained as propagation of charge carriers via evanescent waves (tunneling between the leads) [21]. Under these conditions, the minimum of conductivity can reach the universal minimum value:

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regardless of the edges of the graphene strip. Experimental confirmation of the ballistic transport and the universal minimum conductivity in graphene was provided by low-temperature transport spectroscopy on single-layers and bilayers [22] and through measurements of shot noise at low frequency in field effect devices ( 200 nm long and with

W/L = 24 ) at temperatures around 4.2 K [23]. When the effect of graphene edges cannot be

neglected (W/L<3 ) or at the presence of disorder which locally affects the density of carriers, the evanescent states are accompanied by propagating states and the minimum conductivity rapidly increases [24]. Due to the fabrication process, graphene usually contains various sources of disorder, as defects, impurities, strong interaction with charges in surrounding dielectrics, phonons, etc. This disorder causes spatial inhomogeneities in the carrier density. Local accumulations of charge carriers, so called electron-hole puddles [25], produce percolation paths for carrier transport and prevent the transition to the ideal minimum conductivity state at the Dirac point. Hence, for real graphene the measured conductivity at cryogenic temperatures is much higher than the universal minimum value, changes from sample to sample and is typically in the range 2÷5 e2_{/h on good quality samples [26-28].}

At higher carrier density, i.e. away from the Dirac point, the mentioned disorder sources, acting as scattering centers, reduce the electron mean free path. Two transport regimes are often considered depending on the mean free path length l with respect to the graphene length

L. When l > L, transport is ballistic since carriers can travel through graphene at Fermi

velocity vF without scattering. On the other hand, when l < L, transport is diffusive since

carriers undergo elastic and inelastic collisions. In both cases, transport can be described by the Landauer formalism [29] and the conductivity can be expressed as:

= 2_ℎ ( ) ( ) − , (1.20) with T(E) the transmission function and M(E) the number of conducting channels.

For ballistic transport:

( ) = 1, (1.21) while for diffusive transport:

( ) = _{( ) + , (1.22)}( ) where L is the length of the sample and λ(E) is the energy-dependent scattering mean free path.

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M(E) can be calculated [14] from the dispersion relation eq. (1.8), and, similarly to the density

of states, has a linear dependence on the energy E:

( ) = 2| |_{ħ . (1.23)} From eqs. (1.20)-(1.23), under the approximation that - ∂f/∂E ≈ δ(E-EF) valid for T → 0K, a

simple expression of graphene conductivity can be easily obtained:

=2_ℎ 2_ħ ( ). (1.24) In eq. (1.24), λ = L, independent of the energy E in the ballistic regime, and:

( ) = 2 ( ), (1.25) in the diffusive regime, where τ(E) is the momentum relaxation time, i.e. the average time between scattering events.

Recalling eq. (1.11) and the EF vs. n relation of eq. (1.17), eq. (1.24) implies that ∝ √

and _{∝ 1/√ in the ballistic regime. This dependence, which is sketched in Fig. 1.9, has} been experimentally observed on clean graphene [30].

FIGURE 1.9: Conductivity vs. carrier density (σ vs. n) for graphene. Acoustic phonons (short range) and ionized impurity (long range) scattering are considered [12].

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In the diffusive regime, λ(E) and τ(E) depend on the scattering mechanism. In graphene, three main scattering mechanisms are considered: Coulomb scattering by charged impurities (long range scattering), short-range scattering (defects, adsorbates), and electron-phonon scattering.

Charged impurity scattering is a very important scattering mechanism [30]. It is caused by the presence of charged impurities close to the graphene sheet. These impurities can be trapped ions in the top or bottom dielectric or ions on the graphene surface. Coulomb scattering is more relevant at low energies and the relaxation time corresponding to it varies linearly with energy, τ(E) ∝ E [31,32]. According to eqs. (1.11), (1.17), (1.24) and (1.25),

σ

∝ n and the mobility is independent of n. The observation of a linear

σ vs. n plot (Fig. 1.9)

is frequently taken as evidence for the presence of charged impurity scattering.

Short range scattering potential is due to localized defects as vacancies and cracks [31,33]. The resulting scattering rate is proportional to the final density of states, so 1/τ(E) ∝ E, and is independent of temperature. Hence, for this scattering mechanism, the conductivity does not depend on n and µ ∝ 1/n.

Deformation potential scattering by acoustic phonons [33-35] is another important scattering mechanism. Phonons can be considered an intrinsic scattering source since they limit the mobility at finite temperature even when there are no defects. Longitudinal acoustic (LA) phonons are known to have a higher electron-phonon scattering cross-section. The scattering of electrons by LA phonons can be considered quasi-elastic since the phonon energies are negligible in comparison with the Fermi energy of electrons. Optical phonons in the graphene can also scatter carriers, especially at temperatures above 300 K, and are believed to be responsible for the decrease in conductivity at high temperatures [36]. Phonon scattering is usually invoked to explain the temperature dependence of

σ but it does not

introduce any dependence on n (Fig. 1.9) .

Other scattering mechanisms can affect the conductivity. Different scattering mechanisms add up to produce a total conductivity σTot given by:

1 = 1 + 1 + ⋯, (1.26) where

σ

i is the conductivity corresponding to a given scattering mechanism. According to

eq. (1.26), the smaller σi limits the total σTot. An example is given in Fig. 1.9 where acoustic

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1.2.6 Optical properties

Graphene has remarkable optical properties. The gapless energy band enables charge carrier generation by light absorption over a very wide energy spectrum, unmatched by any other material [12]. This includes the ultraviolet, visible, infrared (IR) and terahertz (THz) spectral regimes. In the near-IR and visible, light transmittance T of graphene does not depend on frequency, being controlled by the fine structure constant

α = e

2_/(4πε

0ћ2c) [37] . At normal

incidence, the transmittance can expressed as [38]:

= (1 − 0.5 ) ≈ 1 − ≈ 0.977. (1.27) Considering the thickness of 0.334 nm, a single-layer of suspended graphene has an unusually high absorption of A = 1-T ≈ 2.3%, corresponding to an absorption coefficient about 50 times higher than for example the absorption of GaAs at λ = 1.55

µ

m and it demonstrates the strong

coupling of light and graphene, which can be exploited for conversion of photons into electrical current or voltage [39,40]. Because graphene sheets behave as a 2-D electron gas, they are optically almost noninteracting in superposition, and the absorbance of few-layer graphene sheets is roughly proportional to the number of layers. The proportionality is gradually lost and the transparency remains quite high while adding further layers: graphene layers corresponding to a thickness of 1

µ

m still have a transparency of approximately 70%

[41].

In addition, the reflectivity of graphene is very low: R = 0.25π2

_α

2_{(1-A) = 1.3×10}-4_{, though}

it increases to 2% for 10 layers [42].

1.3 Carbon nanotubes

Carbon nanotubes (CNTs) are graphitic sheets curled up into seamless cylinders. There are two families of CNTs, namely single-wall CNTs (SWCNTs) and multi-wall CNTs (MWCNTs).A SWCNT (Fig. 1.10(a)) is a hollow cylindrical structure of carbon atoms with a diameter that ranges from about 0.5 to 5 nm and lengths of the order of micrometers to centimeters. An MWCNT (Fig. 1.10(b)) is similar in structure to a SWCNT but has multiple concentric cylindrical walls with the spacing between walls comparable to the interlayer spacing in graphite, approximately 0.34 nm [4].

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FIGURE 1.10: On the left: ball-and-stick models of a SWCNT (a) and a MWCNT (b). The balls represent the carbon atoms and the sticks stand for the bonds between carbon atoms. On the right: high-resolution TEM images of single-wall and multi-wall CNTs observed by Sumio Iijima [44,45].

The large aspect ratio makes the nanotubes nearly ideal one-dimensional (1-D) objects. There are three types of SWCNT: chiral CNTs, armchair CNTs, and zigzag CNTs, of which the latter two are achiral. Depending on the detailed arrangement of the carbon atoms the SWCNTs can be metallic or semiconducting [43]; instead the MWCNTs are usually metallic.

MWCNTs were observed for the first time in transmission electron microscopy (TEM) studies by Iijima in 1991 [44], while SWCNTs were produced independently by Iijima [45] and Bethune [46] in 1993. Fig. 1.10(c) and Fig. 1.10(d) show the TEM images of single-wall and multi-wall CNTs observed by Sumio Iijima.

1.3.1 The direct lattice

To understand the origin of the different types of CNT, we start from the direct lattice of graphene and then define a mathematical construction which folds graphene’s lattice into a CNT. This construction directly leads to a precise determination of the primitive lattice of carbon nanotubes, which is required information in order to derive the CNT band structure.

With reference to Fig. 1.11(a), that shows the honeycomb lattice of graphene, a single-wall CNT can be conceptually conceived by considering folding the dashed line containing primitive lattice points A and C with the dashed line containing primitive lattice points B and

D such that point A coincides with B, and C with D to form the nanotube shown in Fig. 1.11(b) [4].

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FIGURE 1.11: An illustration to describe the conceptual construction of a CNT from graphene. (a)Wrapping or folding the dashed line containing points A and C to the dashed line containing points B and D results in the (3,3) armchair carbon nanotube in (b). The CNT primitive unit cell is the cylinder formed by wrapping line AC onto BD and is also highlighted in (b).

The CNT is characterized by three geometrical parameters, the chiral vector Ch, the

translation vector T, and the chiral angle θ, as shown in Fig. 1.11(a).

Ch is defined as the vector connecting any two primitive lattice points of graphene such

that when folded into a nanotube these two points are coincidental or indistinguishable, and |Ch| is the CNT circumference. Ch is equivalent to:

= + , (1.28) where a1 and a2 are the primitive lattice vectors defined by eq. (1.1), n and m are positive

integers.

The type of CNT can be deduced directly from the values of the chiral vector and it is described as an (n, m) CNT. The (n, n) CNTs are armchair nanotubes, the (n, 0) CNTs are zigzag nanotubes and all other ( n, m) CNTs lead to chiral nanotubes.

The other two geometrical parameters (T and θ) can be derived from the chiral vector. The chiral angle is the angle between the chiral vector and the primitive lattice vector a1:

= _{| || |. (1.29)}∙ Unique values of the chiral angle are restricted to 0° ≤ θ ≤ 30°. All armchair nanotubes have a chiral angle of 30° and θ = 0° for all zigzag nanotubes.

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The translation vector T defines the periodicity of the lattice along the tubular axis and it is the smallest graphene lattice vector perpendicular to Ch.

Let T = t1a1 + t2a2, where t1 and t2 are integers. Therefore:

∙ = (2 + ) + (2 + ) = 0. (1.30) The acceptable solution for eq. (1.30) is:

= 2 + , −2 + , (1.31) where gd is the greatest common divisor of 2m + n and 2n + m.

1.3.2 _{Brillouin zone}

The wavevectors defining the CNT first Brillouin zone are the reciprocals of the primitive unit cell vectors given by the reciprocity condition [4]:

( )∙( ) _{= 1, (1.32)}

where Ka is the reciprocal lattice vector along the nanotube axis and Kc is along the

circumferential direction, both given in terms of the reciprocal lattice basis vectors of graphene defined by eq. (1.5).Eq. (1.32) simplifies to:

∙ = 2 , ∙ = 0, ∙ = 0, ∙ = 2 . (1.33) By the periodic boundary conditions on the Bloch wave functions, the allowed wavevectors k within the Brillouin zone along the axial direction are:

= 2 , = 0,1, … , − 1, (1.34) where Nuc is the number of unit cells in the nanotube of length Lt = NucT. The maximum

integer value of l is determined from the requirement that unique solutions for k are restricted to the first Brillouin zone, i.e. maximum k < |Ka| = 2π/T. In the limit where the CNT is very

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long, for instance Lt ≫T or Nuc ≫1, then the spacing between k-values vanishes and k can be

considered a continuous variable along the axial direction:

= − , , (1.35) where the wavevector has been re-centered to be symmetric about zero consistent with standard Brillouin zone convention.

Applying the periodic boundary conditions to determine the allowed wavevectors q along the circumferential direction yields:

=2 = | |, = 0,1, … , − 1, (1.36) where N is the number of hexagons per unit cell.

We observe that the q-values are separated by a gap that is much greater than the spacing in k-values, i.e. 2π/Ch ≫ 2π/Lt for long CNTs with lengths Lt ≫ Ch. Therefore, the q variable

is quantized or discretely spaced compared with the relatively continuous k variable, which implies that the allowed CNT wavevectors in the Brillouin zone are composed of a series of lines, as shown in Fig. 1.12. These lines are basically 1D cuts of graphene’s reciprocal lattice.

FIGURE 1.12: Brillouin zone of a (3, 3) armchair CNT (shaded rectangle) overlaid on the reciprocal lattice of graphene. The numbers refer to j = 0, 1, ... , 5 for a total of N = 6 1D bands in the CNT Brillouin zone.

Finally, the expression for any arbitrary allowed wavevector within the Brillouin zone is:

= _{2 / +}a _c, = 0,1, … , − 1 and − , (1.37)

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1.3.3 _{Electronic and transport properties}

The electronic structure and electrical properties of SWCNTs are usually discussed in terms of the electronic structure of graphene [4,47]. Fig. 1.15(a) shows the band structure and the hexagonal first Brillouin zone of graphene. The energy bands of CNTs are line cuts or cross-sections of the bands of graphene and the entire Brillouin zone of CNTs can be folded into the first Brillouin zone of graphene. When these cuts pass through a Dirac point, the nanotube is metallic (Fig. 1.13(b)); in cases where no cut passes through a K point, the nanotubes are semiconducting (Fig. 1.13(c)).

FIGURE 1.13: (a) Band structure of a graphene sheet (top) and the first Brillouin zone (bottom). (b) Band structure of a metallic (3,3) CNT. (c) Band structure of a (4,2) semiconducting CNT. The allowed states in the nanotubes are cuts of the graphene bands indicated by the white lines. If the cut passes through a K point, the CNT is metallic; otherwise, the CNT is semiconducting.

It can be shown that an (n,m) CNT is metallic when n = m and when n-m = 3i [48], where i is an integer, while CNTs with n-m≠3i are semiconducting [49,50].

The band structure of CNTs can be computed by inserting the allowed wavevectors, given by eq. (1.37) and rewritten in terms of its and components.

In general, for any (n, m) CNT, there will be N valence bands (E ≤ 0) and N conduction bands (E ≥ 0). Each one of the bands has 2Nuc allowed states, where the factor of 2 is due to

spin degeneracy. At equilibrium the valence bands will be fully occupied and the conduction bands empty with the Fermi energy EF = 0 eV.

Fig. 1.14shows the band structures for (10, 4) metallic and (10, 5) semiconducting chiral CNTs [4].

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FIGURE 1.14: Band structures for (a) (10, 4) metallic CNT and (b) (10, 5) semiconducting CNT, within ±3 eV. The metallic CNT shows a band degeneracy at 0 eV and k = ±2π/3T. The semiconducting CNT has a bandgap of ∼ 0.86 eV.

Fig. 1.15 shows the band structure for an (8,8) armchair CNT, revealing an energy degeneracy at ka = ±2π/3, where the valence band touches the conduction band [4].

FIGURE 1.15: Band structure for (8, 8) armchair nanotube. For all armchair CNTs, the valence band touches the conduction band at ka = ±2π/3, which explains their metallic properties.

In general, the energy degeneracy at 0 eV is common to all armchair CNTs and, hence, armchair CNTs are metallic.

For armchair CNTs, the first subbands of the valence and conduction bands have a linear dispersion at low energies and to a good approximation can be approximated in a simple manner with a linear E−k relation independent of chirality. The linear dispersion for the right-half of the Brillouin zone can be expressed as:

( ) ≈ ћ −2_{3 , 3} , (1.38) where ћ is the reduced Planck’s constant and vF is the Fermi velocity.

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Fig. 1.16 shows the band structures for (12, 0) metallic and (13, 0) semiconducting zigzag CNTs [4].

FIGURE 1.16: Band structures for (a) (12, 0) and (b) (13, 0) zigzag CNTs. The (12, 0) CNT is metallic, while the (13, 0) CNT is semiconducting due to the bandgap at k = 0.

In general, when n is a multiple of 3, the zigzag CNT is metallic, otherwise it is semiconducting.

For metallic zigzag CNTs, a simple linear E−k relation can accurately describe the first subband of the valence and conduction bands. Similar to graphene’s linear dispersion, the linear dispersion for the first subband of metallic zigzag CNTs can be expressed as:

( ) ≈ ћ | |. (1.39) The bandgap for semiconducting CNTs is [4,49]:

≈ 2 , (1.40) where γ is the hopping energy, ac-c is the carbon–carbon bond length and dt is the diameter of

the CNT. Numerically, Eg(eV) ∼ 0.9/dt (nm).

The CNT electronic structure has many 1D subbands; as such, the total DOS gtot at a given

energy is the sum of the contributions from the DOS of each subband [4]:

g ( ) = g( , ). (1.41) where N is the number of subbands in the CNT band structure.

In a 1D solid, the number of states between E and E + dE is the differential wave vector dk normalized to the length of one state:

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g( ) d = 2_{2 / , (1.42)}d where L is the length of the 1D system and 2π/L is the length of one k-state; the factor of 2 in the numerator accounts for spin degeneracy. Hence, it follows that a general formula for the electron DOS in a 1D solid is:

g( ) =1 d_{d . (1.43)} For example, we report the DOS for zigzag (n, 0) nanotubes. Owing to mirror symmetry of the E−k relationship, the DOS for the negative branch of the wavevector is identical to the DOS for the positive branch and the complete DOS for the jth subband is [4]:

g ( , ) = 4 √3

| |

( − )( − ), (1.44) where α accounts for the Brillouin zone mirror symmetry or degeneracy and Evhi (i = 1,2) is

known as a van Hove singularity (VHS) [4].Specifically, α = 1 if E is energy at the Brillouin zone center (since the Г-point center is common to both branches of the wavevector), otherwise α = 2.

The DOS for semiconducting and metallic zigzag nanotubes are shown in Fig. 1.17. A noteworthy insight is that the square of the energy terms in the denominator of the expression for the DOS is due to the electron–hole symmetry present in the NNTB band structure of CNTs leading to mirror symmetry between the conduction and valence bands’ DOS visually evident in Fig. 1.17.

The SWCNTs are 1-D objects and as such their two-terminal conductance is given by Landauer’s equation [50-52]:

= 2_ℎ

, (1.45) where 2e2_{/h is the quantum of conductance, T}

i is the transmission of a contributing

conduction channel (subband), and the sum involves all contributing conduction channels,

i.e., channels whose energylies between the electrochemical potentials of the left andright reservoirs to which the nanotube is connected.

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FIGURE 1.17: The electronic DOS for (a) metallic and (b) semiconducting zigzag nanotubes. For metallic (semiconducting) nanotubes, the DOS around 0 eV is non-zero (zero).

In the absence of any scattering, i.e., when all Ti = 1 , the resistance of a metallic SWCNT is

(h/4e2_{) ≈ 6.5 kΩ, because for the lowest subband in metallic CNTs the number of degenerate}

subbands is Nch = 2 [4]. This quantum mechanical resistance is a contact resistance arising

from the mismatch of the number of conduction channels in the CNT and the macroscopic metal leads.

There is strong evidence that Ti = 1 in the case of metallic SWCNTs, so that these tubes

behave as ballistic conductors [53–56]. This arises from the 1-D confinement of the electrons which allows motion in only two directions. This constraint along with the requirements for energy and momentum conservation severely reduces the phase space for scattering processes. However, in addition to the quantum mechanical contact resistance, there are other sources of contact resistance, such as those produced by the existence of metal-nanotube interface barriers, or poor coupling between the CNT and the leads. These types of resistance are very important and can dominate electrical transport in nanotubes [56].

Unlike SWCNTs, the electrical properties of MWCNTs have received less attention. This is due to their complex structure (every carbon shell can have different electronic character and chirality) and the presence of shell–shell interactions [57,58]. However, at low bias and temperatures, and when MWCNTs are side-bonded to metallic electrodes, transport is dominated by outer-shell conduction [59,60].

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1.4 _{Theory of the field emission}

Controlled propagation of electrons in vacuum is at the basis of several technological applications, like CRT displays, vacuum electronics, electron microscopy, X-ray generation, electron beam lithography, etc. The most common technique to extract electron from matter is thermionic emission, where electrons are emitted from heated filaments (hot cathodes), which requires a source heated at high temperature(~1000 °C) and has several drawbacks. Field emission (FE), which involves extraction of electrons from a conducting solid (metal or highly doped semiconductor) by an external electric field, is becoming one of the best alternatives. Indeed, by this method, an extremely high current density with low energy spread of the emitted electrons and with negligible power consumption can be achieved [61-63].

The phenomenon of field emission is associated with a quantum mechanical tunneling process whereby electrons near the Fermi level tunnel through a (material dependent) potential barrier, whose width is reduced by the application of an external electric field, and escape to the vacuum level (Fig. 1.18) [64].

FIGURE 1.18: Potential-energy diagram illustrating the effect of an external electric field on the energy barrier for electrons at a metal surface [64].

For a parallel flat electrode configuration the field is off the order of 109_{V/m. However, if the}

cathode surface has a high point or a protrusion, electrons may be extracted at a considerably lower applied field [62,65]. This is because the lines of force converge at the sharp point and the physical geometry of the tip provides a field enhancement.

The emission current depends on the electric field at the emitter surface (referred as microscopic or local electric field), ES, and on the workfunction, Φ, i.e. the effective

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surface-vacuum barrier height. The Fowler–Nordheim model [66], derived for a flat metallic surface at 0 K and assuming a triangular potential barrier, predicts an exponential behaviour of the emitted current:

= ∙ − / , (1.46) where S is the emitting surface area, ES is the uniform electric field on that surface and a and

b are constants. When S is expressed in cm2_{, Φ and E}

S respectively in eV and V/cm,

a = 1.54·10-6_A·eV·V-2_{and b = 6.83·10}9 _eV-3/2_·V·m-1_.

In a parallel plate configuration, the field ES can be obtained from the applied potential V

and the inter-electrode distance d as ES = V/d. If the cathode surface has a protrusion, a field

enhancement factor,

β, which takes into account the amplification occurring around the tip,

has to be introduced and:

= . (1.47) According to eqs. (1.46) and (1.47), a Fowler–Nordheim plot of ln(I/V2_{) as a function of 1/V}

is a straight line, whose slope, m = bΦ3/2_d/

_{β, and interception, y}

0 = ln(aSβ2/Φd2), can, in

principle, be used to estimate β and Φ. Although corrections [67,68] are required to describe effects of non-zero temperature, series resistance, extremely curved surfaces and non-uniform field enhancement factors or workfunctions, the basic FN theory has proven to be a good model to achieve a first-approximation understanding of the emission phenomena. For temperatures up to several hundred degree Celsius and fields in a large window, F–N model provides a good fitting to the I–V characteristics of several kind of emitters.

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35

Chapter 2

Graphene-based field-effect transistors

2.1 _Introduction

The Si-based electronics has severe physical limitations for further developments, in particular related to high power dissipation caused by leakage effects. This is especially dangerous when continuing the shrinkage of field effect transistor (FET) dimensions and oxide thicknesses [1,2].

Graphene is a promising candidate for future nanoelectronics. Graphene-based field-effect transistors (GFETs) [3] combine an ultra-thin body suitable for aggressive channel length scaling [4], with excellent properties, as a linear dispersion relation with electrons behaving as massless Dirac fermions [5], a very high carrier mobility [6] and a superior current density capability [7]. In such devices, an electric current is injected/extracted from metallic electrodes (source/drain) through a graphene channel whose conductance is modulated by the electric field from a back- or top-gate. The linear energy dispersion, with zero bandgap and a double-cone shape with intrinsic Fermi level at the vertex, gives symmetric valence and conduction bands; differently from most materials, current modulation by means of a gate in GFETs is possible even without a bandgap, due to the vanishing density of states at the vertex [5, 8].

However the development of graphene-based electronics is limited by the quality of the contacts between the graphene and the metal electrodes [9-11] which can significantly affect the electronic transport properties of the devices [12]. Despite this, the physics of graphene– metal contacts remains still an open subject.

Although the carrier mobility in the graphene is high, the very small DOS for graphene might suppress the current injection from the metal contacts to the graphene, thus resulting in high contact resistivity Rc [10,12]. In particular, a high Rc limits the total on-state current, and

has a severe impact on transistor performance, negatively influencing the peak transconductance as well as the linearity of the current versus gate-voltage characteristic [13]. The use of a four-point setup for electrical characterization is clearly suitable to prevent the problems related to contact resistance, but real electronic applications are based on

Graphene and carbon nanotubes in transistors, diodes and field emission devices

D

IPARTIMENTO DI

F

ISICA “

E

.

R

.

C

AIANIELLO”

THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHYLOSOPHY IN PHYSICS

G

RAPHENE AND

C

ARBON

N

ANOTUBES IN

T

RANSISTORS,

D

IODES AND

F

IELD

E

MISSION

D

EVICES

L

AURA

I

EMMO

S

UPERVISOR

PROF ANTONIO DI BARTOLOMEO

C

OORDINATOR

P

ROF

C

ANIO

N

OCE

Contents

Introduction

Bibliography

Chapter 1

Graphene and carbon nanotubes:

theoretical and experimental background

1.1

Introduction

1.2

Graphene

1.2.1

The direct lattice

1.2.2

The reciprocal lattice

1.2.3

Electronic properties

µ, in

σ and the carrier charge density and is commonly used to characterize the

1.2.4

Graphene nanoribbons

1.2.5

Transport properties

σ should tend to zero, the

σ

σ vs. n plot (Fig. 1.9)

σ but it does not

σ

1.2.6

Optical properties

α = e

µ

µ

α

1.3

Carbon nanotubes

1.3.1

_{Electronic properties}

_α

_{Brillouin zone}

_{Electronic and transport properties}

_{Theory of the field emission}

_{β, and interception, y}

_Introduction